Either each of the sets ( X\[itex]A_\alpha[/itex]) = X , in which case the intersection is all of X, or at least one of them is finite , in which case the intersection is a subset of a finite set and hence finite.

To show 3

Let [itex]A_1,A_2,A_3...A_n \subset X [/itex]be open as X\A is finite or all of X.

To show that [itex] \cap A_{n} \in T [/itex]we must show that [itex]\cap[/itex] X\[itex]A_n[/itex] is either finite or all of X.